2002

From Mathsoc wiki

PY2002 Chaos & Complexity (TP only)

The sole TP-only course in second year. Skimmed through various parts of chaotic and complex behaviour, with applets.

Lecturer: Dr. Stefan Hutzler

Website: Link

As taught in: 2007-2008

1 The Logistic map
2 Period Doubling
3 Feigenbaum Numbers
4 Lyapunov Exponents
5 Damped Driven Pendulum
6 Phase Space
7 Fractals
8 Self Organised Criticality
9 Cellular Automata
10 Pedestrian Dynamics
11 Applets

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The Logistic map

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Period Doubling

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Feigenbaum Numbers

Let $a_1$ be the value of the control parameter for some system at the change from period 1 to period 2 behaviour, $a_2$ be the value of the control parameter at the change from period 2 to period 4 behaviour, and so on. Then we have

$\delta_n = \frac{a_n -a_{n-1}}{a_{n+1} - a_n}$

and this is approximately the same for all $n$ . In the limit,

$\delta = \lim_{n \rightarrow \infty} \delta_n = 4.6692016091$

the Feigenbaum delta.

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Lyapunov Exponents

Lyapunov exponents represent an attempt to measure sensitivity to initial conditions. Consider choosing two different starting values and after every iteration computing the difference between the trajectories:

$| \Delta x_0 |, | \Delta x_1 | , \dots$

One finds numerically that

$\ln |x_n| \propto n$

$\Rightarrow |\Delta x_n | = | \Delta x_0| \exp (\tilde{\lambda} n)$

and $\tilde{\lambda}$ is called the Lyapunov exponent $\lambda$ in the limit $n \rightarrow \infty$ . $\lambda > 0$ indicates exponential divergence of nearby trajectories, a signature of chaos.

So we have

$\tilde{\lambda} = \frac{1}{n} \ln \left| \frac{\Delta x_n}{\Delta x_0} \right|$

and we expand

$\frac{\Delta x_n}{\Delta x_0} = \frac{\Delta x_n}{\Delta x_{n-1}} \frac{\Delta x_{n-1}}{\Delta x_{n-2}} \dots \frac{\Delta x_1}{\Delta x_0}$

$\Rightarrow \tilde{\lambda} = \frac{1}{n} \sum_{i=0}^{n-1} \ln \left|\frac{\Delta x_{i+1}}{\Delta x_i}\right|$

In the infinitesimal limit, $\Delta x_i$ becomes $dx_i$ and as $x_{i+1} = f(x_i)$ we have that

$\frac{\Delta x_{i+1}}{\Delta x_i} = f^{\prime}(x_i)$

thus we have that

$\lambda(x_0) = \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{i=0}^{n-1} \ln \left|f^{\prime} (x_i)\right|$

and we compute $\lambda(x_0)$ over a sample of starting points $x_0$ to obtain the average Lyapunov exponent $\lambda = \frac{1}{N} \sum_i \lambda_i$

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Damped Driven Pendulum

Consider the damped driven pendulum, described by

$mL \ddot{\theta} = -mg \sin \theta - \gamma L \dot{\theta} + A \cos \tilde{\omega}_D t$

To simplify analysis, we write this in dimensionless form by introducting a dimensionless time variable $\tau = t \omega_0 = t \sqrt{\frac{g}{L}}$ . Thus,

$\frac{d}{dt} = \frac{d}{d \tau} \frac{d \tau}{dt} = \omega_0 \frac{d}{dt}$

$\frac{d^2}{dt^2} = \omega_0^2 \frac{d^2}{d \tau^2}$

Our equation of motion becomes

$\frac{d^2 \theta}{d \tau^2} + \sin \theta + \frac{\gamma}{m \omega_0} \frac{d \theta}{d \tau} = \frac{A}{mg} \cos \omega_D \tau$

having divided across by $\omega_0^2 L$ after changing our time variable, and where $\omega_D = \frac{\tilde{\omega}_D}{\omega_0}$

We now define $p = \frac{A}{mg}$ and $\frac{1}{q} = \frac{\gamma}{m \omega_0}$ giving

$\ddot{\theta} + \frac{1}{q} \dot{\theta} + \sin \theta = p \cos \omega_D t$

where we now rename $\tau$ as $t$ . Our control parameters are clearly $p, \frac{1}{q}$ and $\omega_D$ .

This equation may be solved numerically using a method such as the fourth order Runge Kutta method (see SF Computational labs) or the simple Euler method (too crude to actually be used in practice). We introduce new variable $\omega$ and $\Phi$ such that

$\frac{d \Phi}{dt} = \omega_D$

$\frac{d \theta}{dt} = \omega$

$\frac{d \omega}{dt} = - \frac{1}{q} \omega - \sin \theta + p \cos \Phi$

The simple Euler method for solving these equations is given below; essentially it approximates the evolution of a system through a series of finite time steps $\Delta t$ using first order Taylor approximations:

$\Phi(t_0 + \Delta t) = \Phi(t_0) + \frac{d \Phi}{dt} \Bigg|_{t_0} \Delta t = \Phi(t_0) + \omega_D \Delta t$

$\theta(t_0 + \Delta t) = \theta(t_0) + \frac{d \theta}{dt} \Bigg|_{t_0} \Delta t = \theta(t_0) + \omega (t_0) \Delta t$

$\omega(t_0 + \Delta t) = \omega(t_0) + \frac{d \omega}{dt} \Bigg|_{t_0} \Delta t = \omega(t_0) + \Big(- \frac{1}{q} \omega(t_0) - \sin \theta(t_0) + p \cos \Phi(t_0)\Big) \Delta t$

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Phase Space

The pair $x(t)$ and $\dot{x}(t)$ specify the behaviour of a dynamical system (in higher dimensions, these would be vector quantities). We can thus characterise the system at any instant of time by a point in a plot of $\dot{x}(t)$ against $x(t)$ - this is known as phase space.

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Phase Space of the Harmonic Oscillator

Harmonic Oscillator

Referring to our full derivation for the damped driven oscillator above, we now let $\frac{1}{q} = p = \omega_D = 0$ , then

$\dot{\theta} = \omega$

$\dot{\omega} = - \theta$

hence

$\frac{\dot{\theta}}{\dot{\omega}} = \frac{\frac{d \omega}{dt}}{\frac{d \theta}{d dt}} = \frac{d \omega}{d \theta} = - \frac{\theta}{\omega}$

$\int \omega d \omega = - \int \theta d \theta$

$\Rightarrow \omega^2 + \theta^2 = C$

the equation of a circle. Note that this derivation used the dimensionless form of the variables, otherwise we would obtain $\omega^2 + \omega_0^2 \theta^2 = C$ , an ellipse. Here the constant $C$ is proportional to the total energy of the oscillator.

Damped Harmonic Oscillator

Here $\frac{1}{q} > 0$ and $p = \omega_D = 0$ . We can determine the fixed points easily:

$\frac{d \theta}{dt} = \omega = 0 \Rightarrow \omega = 0$

$\frac{d \omega}{dt} = 0 = - \frac{\omega}{q} - \theta = 0 \Rightarrow \theta = 0$

so $(0,0)$ is an attractor. The basin of attraction is here the whole plane.

Non-linear pendulum

We have

$\frac{d \omega}{d \theta} = - \frac{\sin \theta}{\omega}$

and integrating gives

$\frac{\omega^2}{2} - \cos \theta = C$

We determine the fixed points as before

$\frac{d \theta}{dt} = \omega = 0 \Rightarrow \omega = 0$

$\frac{d \omega}{dt} = 0 = - sin \theta = 0 \Rightarrow \theta = 0, \pm \pi, \pm 2 \pi \dots$

We find that $\theta = 0, \pm 2 \pi \dots$ are stable and $\theta = \pm \pi, \pm 3 \pi\dots$ are unstable.

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Properties of Phase Space

Trajectories cannot cross as this would violate determinism.

Conservative systems preserve area in phase space, while dissipative ones lead to a decrease in area. Consider a system of the form

$\dot{x}_1 = f_1(x_1, \dots x_n)$

$\vdots$

$\dot{x}_n = f_n(x_1, \dots x_n)$

then the test for dissipation is

$\vec{\nabla} \cdot \vec{f} < 0$

If $\vec{\nabla} \cdot \vec{f} = 0$ , then the system is conservative.

In one-dimension the fixed points divide the x-axis into a number of non-interacting regions, Fixed points can be nodes, repellors or saddle points.

In 2-dimensions, the Poincare-Bendixson theorem states that given that the long-term motion of a system in 2-d space is limited to a finite size region $R$ , and that any trajectory starting in $R$ ends in $R$ , then any trajectory starting in $R$ can only approach a fixed point or limit cycle as $t\rightarrow \infty$ (ie no chaos in 2-d).

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Stability of Fixed Points

In one-dimension, we have

$\lambda = \frac{df}{dx}\Big|_{x_0}$

and if $\lambda > 0$ the fixed point $x_0$ is a repellor, and if $\lambda < 0$ the fixed point $x_0$ is a node.

In two-dimensions, we construct the Jacobi matrix of partial derivatives

$\begin{pmatrix} \frac{df_1}{dx_1} & \frac{df_1}{dx_2} \\ \frac{df_2}{dx_1} & \frac{df_2}{dx_2} \end{pmatrix}$

evaluate it at the fixed point and then find the eigenvalues, which determine the stability of the fixed points.

$\lambda_1 \geq \lambda_2 \geq 0$ - repellor

$\lambda_1 \leq \lambda_2 \leq 0$ - node

$\lambda_1 < 0 < \lambda_2$ - saddle

If the eigenvalues are complex, then the fixed point is at the centre of a spiral (spiralling inwards if the real parts are negative).

The three-dimensional analysis is similar.

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Fractals

A fractal is some geometric object having non-integer dimension.

We can compute fractal dimension as follows: if $\lim_{l\rightarrow 0} N(l) l = \infty$ and there exists a number $D_f$ such that

$\lim_{l \rightarrow 0} N(l) l^{D_f} = B < \infty$

then $D_f$ is called the fractal dimension of the object in question. We have

$\lim_{l \rightarrow 0} l^{D_f} = \lim_{l \rightarrow 0} \frac{B}{N(l)}$

$\Rightarrow \lim_{l \rightarrow 0} D_f \ln l = \lim_{l \rightarrow 0} \left( \ln B - \ln N(l)\right)$

$\Rightarrow D_f = \lim_{l \rightarrow 0} \frac{\ln B}{\ln l} - \lim_{l \rightarrow 0} \frac{\ln N(l)}{\ln l}$

and the first term goes to zero as $\ln l \rightarrow - \infty$ as $l \rightarrow 0$ , thus

$D_f = \lim_{l \rightarrow 0} \frac{\ln N(l)}{\ln (\frac{1}{l})}$

An alternate method is the box counting method, in which one covers the boundary of the object with boxes (or d-spheres in higher dimensions). If $N(l)$ is the number of boxes of side length $l$ needed to cover the boundary, and if

$N(l) \propto l^{-D}$ for $l \rightarrow 0$ (ie if a graph of log (number of boxes) against log (1/sidelength of boxes) is a straight line with slope $D$ )